BUSI 721:
Data Driven Finance I


Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

Daily Returns

Let’s compute close-to-close daily stock returns.

From close one day to close the next.

Let Pt denote the price at close on day t.

If there are no dividends, then the gain on day t is \(P_t - P_{t-1}\).

The return per $1 invested in a share at close on t-1 is \[ \frac{P_t~-P_{t-1}}{P_{t-1}}\ = \frac{P_t}{P_{t-1}} -1 \]

What about dividends?  A Chevron announcement

https://www.chevron.com/newsroom/2021/Q3/chevron-announces-2Q-2021-quarterly-dividend

Nasdaq’s statement that appeared Aug 17, 2021

Chevron Corporation (CVX) will begin trading ex-dividend on August 18, 2021. A cash dividend payment of $1.34 per share is scheduled to be paid on September 10, 2021. Shareholders who purchased CVX prior to the ex-dividend date are eligible for the cash dividend payment.

Three Dates:
August 18: (begins trading ex-dividend)
August 19: (shareholders of record receive the dividend)
September 10: (dividend is paid)

Ex-Dividend Days

If you buy on Aug 18, you are not entitled to the dividend.

Purchase Aug 17 or before (and hold through close Aug 17) → will be shareholder of record on Aug 19
→ receive dividend on Sept 10

Why Aug 17 → Aug 19?

T+2 settlement

How to compute daily returns in August?
Where does the dividend go? (Or does it go in September?)

Treat the dividend
as if it were paid on the ex date (Aug 18)

Aug 18 return = return from close Aug 17 to close Aug 18 is

\[ \frac{1.34+P_{Aug18}-P_{Aug17}}{P_{Aug17}}\ \]

August Return

In Module 1, we discussed compounding. An investment starting from $1 grows over a period of time to

\[ (1+r_1)(1+r_2)...(1+r_N) \]

The rate of return over that period as

\[ (1+r_1)(1+r_2)...(1+r_N)-1 \]

Likewise, we calculate the Aug return for CVX as the product of 1 + daily returns (minus 1).

If there were no dividends, then the return on each day t is

\[ r_t = \frac{P_{t}}{P_{t-1}}-1 → 1 + r_t = \frac{P_{t}}{P_{t-1}}\ \]

                                 →

\[ (1+r_1)(1+r_2)...(1+r_{N-1})(1+r_N)=\frac{P_{1}}{P_{0}}\frac{P_{2}}{P_{1}}...\frac{P_{N-1}}{P_{N-2}}\frac{P_{N}}{P_{N-1}}=\frac{P_{N}}{P_{0}} \]

Interpret 0 = last trading day of July,

N = last trading day of Aug.

Because Chevron went ex-dividend on Aug 18, 1 + Aug 18 return is the sum of two terms

\[ \frac{1.34}{P_{Aug17}} and \frac{P_{Aug18}}{P_{Aug17}} \]

Therefore the product of 1 + returns is the sum of two terms. The term involving the dividend 1.34 simplifies to

\[ \frac{1.34 x \frac{P_N}{P_{Aug18}}}{P_{0}} \]

The numerator is what you would have if you received the dividend on Aug 18, invested it at the close in new shares, and held until the end of Aug.

Conclusion

Daily return treats the dividend as being received on the ex-dividend day even though it is not received until later.

Monthly return from compounding daily returns treats the dividend as being received on the ex-dividend day and reinvested in new shares on that day.

Annual returns are computed the same way - by compounding close-to-close daily returns.

Split-Adjusted Prices

Data vendors routinely compute split-adjusted prices.

If a company does a 2-for-1 share split, then each shareholder gets 1 new share for each of her existing shares. Shares are worth roughly half as much.

Vendors scale down old prices in the same ratio for comparability to new prices.

Yahoo’s Adjusted Prices

finance.yahoo.com is a good source for data, and there are python libraries that interface with its API.

Yahoo’s adjusted closing prices are adjusted for splits and also adjusted for dividends on each ex date.

On Aug 18, 2021, the Aug 17 price was adjusted as

\[ {\hat{P}_{Aug17}}={P_{Aug17}}-1.34 \]

The idea is that 1.34 of the value of CVX at close on Aug 17 was the value of the dividend.

The adjusted price was the value on Aug 17 without right to the dividend.

All prior adjusted prices were re-adjusted on Aug 18 by the factor \[ {\hat{P}_{Aug17}}/{P_{Aug17}} \]

The result is that on days t are not ex days, \[ \frac{\hat{P}_{t}}{\hat{P}_{t-1}}=\frac{P_{t}}{P_{t-1}}=1+r_t \]

On ex days,

\[ \frac{\hat{P}_{t}}{\hat{P}_{t-1}}=\frac{P_{t}}{P_{t-1}-D_t} \]

\[ \frac{\hat{P}_{t}}{\hat{P}_{t-1}}=\frac{P_{t}}{P_{t-1}-D_t}≈\frac{P_{t}+D_t}{P_{t-1}} \]

Example: CVX on Aug 18, 2021:

\[ \frac{P_{t}}{P_{t-1}-D_t}=-0.0271 \]

\[ \frac{P_{t}+D_t}{P_{t-1}}=-0.0267 \]

Getting Returns from Yahoo

!pip install --upgrade pandas-datareader

from pandas_datareader import DataReader as pdr
price = pdr('cvx', 'yahoo', start=2010)['Adj Close']

ret_daily = price.pct_change()

ret_monthly = price.resample('M').last().pct_change()
ret_monthly.index = ret_monthly.index.to_period('M')

ret_annual = price.resample('Y').last().pct_change()
ret_annual.index = ret_annual.index.to_period('Y')

Comments

1. % change in monthly or annual yahoo-adjusted closing prices is equivalent to compounding yahoo daily returns.

2. pdr retrieves open, high, low, close, and volume columns too. We just kept the adjusted close.

3. The data frame created by pdr is indexed by dates in datetime format.

4. Converting the dates for monthly and annual returns to period format is optional.

5. You can input multiple tickers in a list to pdr. You get a data frame with multi-indexed columns, tickers being the inside index. The same code produces data frames of returns with tickers being columns.

6. pdr works with other data sources too, including Federal Reserve Economic Data.